This patent relates to the field of processes for processing acoustical or electromagnetic signals, especially in inhomogeneous media. Since the acoustic and electromagnetic cases are for this invention, interchangeable, the acoustic case shall be used primarily herein for illustration and explanation. It will be apparent how the invention extends into other field phenomenon problems.
Such analysis of acoustical information is one of a number of problems of the class involving the calculation of a wave potential within a region based upon detected conditions at points within that region or at the boundaries of the region. Such problems include, but are not restricted to seismic, acoustic, radar field imaging and similar techniques; they are extremely important in the field of geophysical surveys which image an area of interest using arrays of acoustic, vibrational, magnetic, electrical, nuclear, or gravitational sensors. In each such case, the media through which the signal of interest propagates is naturally inhomogeneous, and the problem arises of predicting the desired signal, in the form of a field potential, at discrete sampling points (the sensor locations) given the inhomogeneous media.
It is typical that such sensor information is analyzed by detecting and displaying various acoustic or other velocity fields across the area.
It is known that the equation describing an acoustical velocity potential, the Helmholtz equation .gradient..sup.2 u+k.sup.2 u=0 where u is the velocity potential and k is the wave number, could be solved in terms of boundary integral equations by the use of Green's theorems. However, such solutions are indeterminant at values of the wave number corresponding to the periods of the possible modes of simple harmonic vibration which may take place within a closed rigid envelope having the form of the boundary surface of the area being analyzed. Therefore, while the introduction of solid state electronic computers and associated numerical analysis techniques have made it feasible to analyze classical boundary integral equations, problems still exist in the area of avoiding non-uniqueness and non-existence problems and in the overall computational complexity of the chosen numerical techniques which may require extensive equipment such as supercomputers or extended periods of time for analysis even on the fastest available computers.
Wave propagation problems in inhomogeneous media, where the wave propagation velocity is variable in two or more directions, are presently solved almost exclusively using various variations and combinations of finite difference and finite element methods. Well-known papers describe finite difference methods and finite element methods for wave propagation problems. The literature for both of these methods is vast and deep. The solution of practical wave propagation problems using finite differences or finite elements generally involves the use of a supercomputer.
For certain frequency ranges and geometry restrictions, other methods exist that take advantage of simplifying assumptions. For example, "high" frequency waves in "gradually" varying media can often be approximated using a variety of "ray theoretical" methods. If the inhomogeneity is in only one direction, multitudes of other "range invariant" methods can be used. For many problems where the largest part of the media's wave propagation speed variation occurs in one coordinate direction, such as in underwater acoustics, range invariant techniques are still used, even if the results are less than ideal. Range invariant methods generally require a workstation computer (a few MFLOPS) to find solutions to practical problems in reasonable times (tens of minutes to hours). Ray theoretic methods, when they can be used, often require less computing power.